By Enrico Arbarello, Corrado De Concini (auth.), Eduard Casas-Alvero, Gerald Welters, Sebastian Xambó-Descamps (eds.)
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Additional info for Algebraic Geometry Sitges (Barcelona) 1983: Proceedings of a Conference held in Sitges (Barcelona), Spain October 5–12, 1983
L e t F a generic transversal a generic even to D even P r o o f . Let be n only a reduced invevtible and let invevtible of i n t e r s e c t i o n of We D analogous be D be degree the union of D even on D . of D G on and of let and C multidegree, Let Q. ,p2 n and a there conic exists G®O D - F(1). be t h e points Q. the u s e d i n t h e p r o o f of p r o p . cuvve thetachavactevistic C with We h a v e t h e e x a c t plane thetachavactevistic the sketch results. argument, 13.
And passing (the 2:1 images Enriques very defines a g:l cover over C~ surface a system a so-called Reye-congruence of an are chosen in belong to the closure of weights smooth a h2 in some elliptic fibration 12Fil on X. C2=i0o as fi hI 3~i<9. IC71 P2 [Ve]. maps doubly X through birationally onto six edges of a the of F8,F9,FIo,F~,F~,FI'o). surface is the one by which This realization Enriques in 1984 discove- red the surfaces n o w bearing his name. i = 8: C 8 = F9+FIo, is IC81 a pencil of hyperelliptic curves (g=2) with double cover in P4" This two base points.
O i z one-parameter and very a nodal observes that s o l u t i o n s of t h i s Euler's curve from the two l i n e s Q: x ° + x 3 = x I + x 2 = O P: x ° + x 2 = x I + x 3 = O by a method he c a l l e d the T - t r a n s f o r m . Let not on P and Q. ting x be or two Q of this y X L one the is the skew lines on there is exactly In general with nimality course P,Q It w o r k s a s f o l l o w s : a one will intersect obtains involution covering smooth line X quartic L involution Given through x extends for the T p ~~ regularly double to quadric x6X meeting in a fourth point a birational involution TpQ X.
Algebraic Geometry Sitges (Barcelona) 1983: Proceedings of a Conference held in Sitges (Barcelona), Spain October 5–12, 1983 by Enrico Arbarello, Corrado De Concini (auth.), Eduard Casas-Alvero, Gerald Welters, Sebastian Xambó-Descamps (eds.)